TY - JOUR
T1 - Distribution of lattice orbits on homogeneous varieties
AU - Gorodnik, Alex
AU - Weiss, Barak
N1 - Funding Information:
1.6 Acknowledgements. We thank Amos Nevo and Ralf Spatzier for many useful and insightful conversations. Thanks also to Hee Oh, Dave Morris and Michael Levin. We thank the Center for Advanced Studies in Mathematics at Ben-Gurion University for funding Gorodnik’s visit to Be’er Sheva, when this work was conceived, and the Technion’s hospitality, which made additional progress possible. The first author is partially supported by NSF grant 0400631, and the second by the Israel Science Foundation.
PY - 2007/4/1
Y1 - 2007/4/1
N2 - Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H G. For an increasing family of finite subsets > 0, a dense orbit υ• Γ, υ V and compactly supported function φ on V, we consider the sums S (T) = ΓT. Understanding the asymptotic behavior of S φ,υ (T) is a delicate problem which has only been considered for certain very special choices of H,G and Γ T. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have dg, where G T = g G: ||g|| < T and Γ T = G T Γ. We also show that the asymptotics of S φ, υ (T) is governed by where ν is an explicit limiting density depending on the choice of υ and || • ||.
AB - Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H G. For an increasing family of finite subsets > 0, a dense orbit υ• Γ, υ V and compactly supported function φ on V, we consider the sums S (T) = ΓT. Understanding the asymptotic behavior of S φ,υ (T) is a delicate problem which has only been considered for certain very special choices of H,G and Γ T. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have dg, where G T = g G: ||g|| < T and Γ T = G T Γ. We also show that the asymptotics of S φ, υ (T) is governed by where ν is an explicit limiting density depending on the choice of υ and || • ||.
KW - Equidistribution
KW - Lattices in Lie groups
KW - Values of quadratic forms
KW - Volume asymptotics
UR - http://www.scopus.com/inward/record.url?scp=34247635165&partnerID=8YFLogxK
U2 - 10.1007/s00039-006-0583-6
DO - 10.1007/s00039-006-0583-6
M3 - Article
AN - SCOPUS:34247635165
SN - 1016-443X
VL - 17
SP - 58
EP - 115
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 1
ER -